text: The reciprocal theorem describes the relationship between two Stokes flows in the same geometry where inertial effects are insignificant compared to viscous effects.  Consider a fluid filled region  bounded by surface  with a unit normal .  Suppose we have solutions to Stokes equations in the domain  possessing the form of the velocity fields  and . The velocity fields harbor corresponding stress fields  and  respectively. Then the following equality holds:
  The reciprocal theorem allows us to obtain information about a certain flow by using information from another flow. This is preferable to solving Stokes equations, which is difficult due to not having a known boundary condition. This particularly useful if one wants to understand flow from a complicated problem by studying the flow of a simpler problem in the same geometry.
  One can use the reciprocal theorem to relate the swimming velocity, , of a swimmer subject to a force  to its swimming gait :
  Now that we have established that the relationship between the instantaneous swimming velocity in the direction of the force acting on the body and its swimming gait follow the general form
  where  and  denote the positions of points
  on the surface of the swimmer, we can establish that locomotion is independent of rate. Consider a swimmer that deforms in a periodic fashion through a sequence of motions between the times  and  The net displacement of the swimmer is
  Now consider the swimmer deforming in the same manner but at a different rate. We describe this with the mapping
  Using this mapping, we see that
  This result means that the net distance traveled by the swimmer does not depend on the rate
  at which it is being deformed, but only on the geometrical sequence of shape. This is the first key result.